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OPEN
received: 09 April 2015 accepted: 01 October 2015 Published: 02 November 2015
Transition and Damping of Collective Modes in a Trapped Fermi Gas between BCS and Unitary Limits near the Phase Transition Hang Dong, Wenyuan Zhang, Li Zhou & Yongli Ma We investigate the transition and damping of low-energy collective modes in a trapped unitary Fermi gas by solving the Boltzmann-Vlasov kinetic equation in a scaled form, which is combined with both the T-matrix fluctuation theory in normal phase and the mean-field theory in order phase. In order to connect the microscopic and kinetic descriptions of many-body Feshbach scattering, we adopt a phenomenological two-fluid physical approach, and derive the coupling constants in the order phase. By solving the Boltzmann-Vlasov steady-state equation in a variational form, we calculate two viscous relaxation rates with the collision probabilities of fermion’s scattering including fermions in the normal fluid and fermion pairs in the superfluid. Additionally, by considering the pairing and depairing of fermions, we get results of the frequency and damping of collective modes versus temperature and s-wave scattering length. Our theoretical results are in a remarkable agreement with the experimental data, particularly for the sharp transition between collisionless and hydrodynamic behaviour and strong damping between BCS and unitary limits near the phase transition. The sharp transition originates from the maximum of viscous relaxation rate caused by fermion-fermion pair collision at the phase transition point when the fermion depair, while the strong damping due to the fast varying of the frequency of collective modes from BCS limit to unitary limit.
Strongly-interacting two-component Fermi gases provide a unique testing ground for the many-body theories of exotic systems, such as unconventional superconductors, nuclear matter, neutron stars and dilute atomic Fermi gases, which at first sight have tunable and strong interactions by using a Feshbach resonance (FR)1,2. Near the resonance [η ~ 0, as usual, we conveniently measure the interaction strength in terms of the inverse scattering length η = (kFasc)−1, where kF is the Fermi momentum and asc the s-wave scatting length], the interparticle interactions are unitary limited and universal1–4. The study of collective excitations in these systems has attracted much attention in the past decades. The collective excitations are one of the main sources to prob the dynamics of the many-body systems. The high accuracy of frequency measurements and the sensitivity of collective phenomena to interaction effects make them good candidates to unravel the dynamical correlations. Experimental results have been obtained for low-lying collective modes of a two-component Fermi gas 6Li in wide temperature and interaction regimes, including the radial compression modes5–10, the axial compressions modes8,11,12, the radial quadrupole modes10,13,14, and the scissors mode10,13. The thermodynamic quantities like energy and entropy in trapped Fermi gases at unitarity are measured without invoking any specific theoretical model15,16. These experiments have in turn stimulated a considerable amount of theoretical works11,12,17–22. Surface Physics Laboratory and Department of Physics, Furan University, Shanghai 200433, China. Correspondence and requests for materials should be addressed to Y.M. (email: [email protected]) Scientific Reports | 5:15848 | DOI: 10.1038/srep15848
1
www.nature.com/scientificreports/ However, a theoretical description of this unitary regime is still challenging, particularly at nonzero temperature for sharp transition and strong damping of the collective modes. There are different strong-coupling theories to study the collective excitations of superfluid Fermi gases in the BCS-BEC crossover. One of them is microscopic theory based on a model Hamiltonian either with a one-channel model for a broad (or weak) FR or with a two-channel model for a narrow FR. The link of these two models is well described in23. There are numerous efforts to develop the strong-coupling perturbation theories of interacting fermions. For example, the thermodynamic potential (or action) approach24–28, the diagrammatic method29–39, and the many-body T-matrix fluctuation theories29–44. Leggett’s45 mean-field theory and then Randeria et al.46 by adding fluctuations get some qualitative correct results at zero temperature. The Quantum Monte Carlo (QMC) simulations47,48 and the pseudogap approach43,49–51 have better results in the BCS-BEC crossover. Because the strong coupling atomic Fermi gases are trapped in a finite space at finite temperatures, the inhomogeneous feature of the system, strong pairing fluctuations, and finite temperatures are important keys in considering real cold Fermi gases, which makes the pure microscopic approach difficult to deal with, especially in studying the collective excitations. Another viewpoint is the quantum hydrodynamical theory based both on the Boltzmann-Vlasov kinetic equation in the normal state17–22 and on the generalized Gross-Pitaevskii(GP) equation in the superfluid state52–62 with a phenomenological equation of state. The study of the viscosity of strongly interacting systems is also a topic of great interest both in experimental works5–14,63,64 and in theoretical works based on the Boltzmann-Vlasov kinetic equation17–22,65,66. The radial compression mode reveals a surprising behavior8: An abrupt change of the radial collective frequency in a strongly attractive Fermi gas. The radial quadrupole mode has confirmed the transition from collisionless to hydrodynamic behavior at η − 0.814. The transition is accompanied by very strong damping. The corresponding features cannot be explained on the basis of available theoretical models and new physics is in great need in this regime. How to explain this feature is an open question by now. We still lack a full discussion on the transition and damping of collective modes, especially compared with the experimental results5–14,63,64. This is the major motivation for our present study of different collective modes under similar experimental conditions where the system is trapped around the critical temperature Tc. In this paper, we determine the sharp transition and strong damping of the collective modes at − 2
OPEN
received: 09 April 2015 accepted: 01 October 2015 Published: 02 November 2015
Transition and Damping of Collective Modes in a Trapped Fermi Gas between BCS and Unitary Limits near the Phase Transition Hang Dong, Wenyuan Zhang, Li Zhou & Yongli Ma We investigate the transition and damping of low-energy collective modes in a trapped unitary Fermi gas by solving the Boltzmann-Vlasov kinetic equation in a scaled form, which is combined with both the T-matrix fluctuation theory in normal phase and the mean-field theory in order phase. In order to connect the microscopic and kinetic descriptions of many-body Feshbach scattering, we adopt a phenomenological two-fluid physical approach, and derive the coupling constants in the order phase. By solving the Boltzmann-Vlasov steady-state equation in a variational form, we calculate two viscous relaxation rates with the collision probabilities of fermion’s scattering including fermions in the normal fluid and fermion pairs in the superfluid. Additionally, by considering the pairing and depairing of fermions, we get results of the frequency and damping of collective modes versus temperature and s-wave scattering length. Our theoretical results are in a remarkable agreement with the experimental data, particularly for the sharp transition between collisionless and hydrodynamic behaviour and strong damping between BCS and unitary limits near the phase transition. The sharp transition originates from the maximum of viscous relaxation rate caused by fermion-fermion pair collision at the phase transition point when the fermion depair, while the strong damping due to the fast varying of the frequency of collective modes from BCS limit to unitary limit.
Strongly-interacting two-component Fermi gases provide a unique testing ground for the many-body theories of exotic systems, such as unconventional superconductors, nuclear matter, neutron stars and dilute atomic Fermi gases, which at first sight have tunable and strong interactions by using a Feshbach resonance (FR)1,2. Near the resonance [η ~ 0, as usual, we conveniently measure the interaction strength in terms of the inverse scattering length η = (kFasc)−1, where kF is the Fermi momentum and asc the s-wave scatting length], the interparticle interactions are unitary limited and universal1–4. The study of collective excitations in these systems has attracted much attention in the past decades. The collective excitations are one of the main sources to prob the dynamics of the many-body systems. The high accuracy of frequency measurements and the sensitivity of collective phenomena to interaction effects make them good candidates to unravel the dynamical correlations. Experimental results have been obtained for low-lying collective modes of a two-component Fermi gas 6Li in wide temperature and interaction regimes, including the radial compression modes5–10, the axial compressions modes8,11,12, the radial quadrupole modes10,13,14, and the scissors mode10,13. The thermodynamic quantities like energy and entropy in trapped Fermi gases at unitarity are measured without invoking any specific theoretical model15,16. These experiments have in turn stimulated a considerable amount of theoretical works11,12,17–22. Surface Physics Laboratory and Department of Physics, Furan University, Shanghai 200433, China. Correspondence and requests for materials should be addressed to Y.M. (email: [email protected]) Scientific Reports | 5:15848 | DOI: 10.1038/srep15848
1
www.nature.com/scientificreports/ However, a theoretical description of this unitary regime is still challenging, particularly at nonzero temperature for sharp transition and strong damping of the collective modes. There are different strong-coupling theories to study the collective excitations of superfluid Fermi gases in the BCS-BEC crossover. One of them is microscopic theory based on a model Hamiltonian either with a one-channel model for a broad (or weak) FR or with a two-channel model for a narrow FR. The link of these two models is well described in23. There are numerous efforts to develop the strong-coupling perturbation theories of interacting fermions. For example, the thermodynamic potential (or action) approach24–28, the diagrammatic method29–39, and the many-body T-matrix fluctuation theories29–44. Leggett’s45 mean-field theory and then Randeria et al.46 by adding fluctuations get some qualitative correct results at zero temperature. The Quantum Monte Carlo (QMC) simulations47,48 and the pseudogap approach43,49–51 have better results in the BCS-BEC crossover. Because the strong coupling atomic Fermi gases are trapped in a finite space at finite temperatures, the inhomogeneous feature of the system, strong pairing fluctuations, and finite temperatures are important keys in considering real cold Fermi gases, which makes the pure microscopic approach difficult to deal with, especially in studying the collective excitations. Another viewpoint is the quantum hydrodynamical theory based both on the Boltzmann-Vlasov kinetic equation in the normal state17–22 and on the generalized Gross-Pitaevskii(GP) equation in the superfluid state52–62 with a phenomenological equation of state. The study of the viscosity of strongly interacting systems is also a topic of great interest both in experimental works5–14,63,64 and in theoretical works based on the Boltzmann-Vlasov kinetic equation17–22,65,66. The radial compression mode reveals a surprising behavior8: An abrupt change of the radial collective frequency in a strongly attractive Fermi gas. The radial quadrupole mode has confirmed the transition from collisionless to hydrodynamic behavior at η − 0.814. The transition is accompanied by very strong damping. The corresponding features cannot be explained on the basis of available theoretical models and new physics is in great need in this regime. How to explain this feature is an open question by now. We still lack a full discussion on the transition and damping of collective modes, especially compared with the experimental results5–14,63,64. This is the major motivation for our present study of different collective modes under similar experimental conditions where the system is trapped around the critical temperature Tc. In this paper, we determine the sharp transition and strong damping of the collective modes at − 2
